边界条件含有谱参数的不连续二阶微分算子的$J$-自伴性和格林函数

数学物理学报 ›› 2025, Vol. 45 ›› Issue (3): 707-725.

边界条件含有谱参数的不连续二阶微分算子的$J$-自伴性和格林函数

付慧洁(),许美珍^*()

内蒙古工业大学理学院呼和浩特 010051

收稿日期:2024-07-19 修回日期:2025-01-30 出版日期:2025-06-26 发布日期:2025-06-20
通讯作者: *许美珍， E-mail: xumeizhen1969@163.com
作者简介:付慧洁， E-mail: fuhuijie1008@163.com
基金资助:
国家自然科学基金(12261066);内蒙古自然科学基金(2023LHMS01015);内蒙古自治区直属高校基本科研业务费(JY20240043)

$J$-Self-Adjointness and Green's Function of Discontinuous Second-Order Differential Operator with Eigenparameters in the Boundary Conditions

Fu Huijie(),Xu Meizhen^*()

College of Sciences, Inner Mongolia University of Technology, Hohhot 010051

Received:2024-07-19 Revised:2025-01-30 Online:2025-06-26 Published:2025-06-20
Supported by:
NSFC(12261066);NSF of Inner Mongolia(2023LHMS01015);Basic Science Research Fund of the Universities Directly under the Inner Monngolia Autonomous Region(JY20240043)

摘要/Abstract

摘要：

考虑一类边界条件两端含有谱参数且具有转移条件的二阶复系数微分算子的 $J$-自伴性和格林函数. 通过在适当的 Hilbert 空间上定义一个与问题相关的线性算子 $A$, 将所研究的问题转化为对此空间中算子的研究, 并证明算子 $A$ 是 $J$-自伴的. 另外, 给出算子的基本解和基本解的渐近式, 进一步, 讨论算子的格林函数和预解算子, 并给出格林函数的渐近式.

关键词: 二阶复系数微分算子, 边界条件, 谱参数, $J$-自伴, 格林函数

Abstract:

In this paper, the $J$-self-adjointness and Green's function of a class of discontinuous second-order complex coefficient differential operator with eigenparameters in the boundary conditions of both endpoints are considered. By introducing a linear operator $A$ related to the problem in a suitable Hilbert space, the considered problem can be interpreted as the study of the operator in this space, and it is proved that the operator $A$ is $J$-self-adjoint. In addition, the basic solutions of the operator and the asymptotic formula of the basic solutions are given. Furthermore, the Green's function and the resolvent operator of this operator are derived, and the asymptotic formula of Green's function is given.

Key words: second-order complex coefficient differential operator, boundary conditions, eigenparameters, $J$-self-adjointness, Green's function

中图分类号:

O175.3

付慧洁, 许美珍. 边界条件含有谱参数的不连续二阶微分算子的$J$-自伴性和格林函数[J]. 数学物理学报, 2025, 45(3): 707-725.

Fu Huijie, Xu Meizhen. $J$-Self-Adjointness and Green's Function of Discontinuous Second-Order Differential Operator with Eigenparameters in the Boundary Conditions[J]. Acta mathematica scientia,Series A, 2025, 45(3): 707-725.

参考文献 21

[1]	Mukhtarov O Sh, Kadakal M. Some spectral properties of one Sturm-Liouville type problem with discontinuous weight. Sib Math J, 2005, 46(4): 681-694
[2]	Akdoğan Z, Demirci M, Mukhtarov O Sh. Green function of discontinuous boundary value problem with transmission conditions. Math Models Methods Appl Sci, 2007, 30(14): 1719-1738
[3]	李昆, 郑召文. 一类具有转移条件的 Sturm-Liouville 方程的谱性质. 数学物理学报, 2015, 35A(5): 910-926
	Li K, Zheng Z W. Spectral properties for Sturm-Liouville equations with transmission conditions. Acta Math Sci, 2015, 35A(5): 910-926
[4]	Zhang X Y, Sun J. Green function of fourth-order differential operator with eigenparameter-dependent boundary and transmission conditions. Acta Math Appl Sin, 2017, 33(2): 311-326
[5]	Cai J M, Zheng Z W, Li K. A class of singular Sturm-Liouville problems with discontinuity and an eigenparameter-dependent boundary condition. Mathematics, 2022, 10( 23): 4430
[6]	Ozkan A S, Keskin B. Spectral problems for Sturm-Liouville operator with boundary and jump conditions linearly dependent on the eigenparameter. Inverse Probl Sci Eng, 2012, 20(6): 799-808
[7]	Manafov M D. Inverse spectral problems for energy-dependent Sturm-Liouville equations with finitely many point $\delta$-interactions. Electron J Differ Equ Conf, 2016, 2016(11): 1-12
[8]	Liu Y X, Shi G L, Yan J. An inverse problem for non-selfadjoint Sturm-Liouville operator with discontinuity conditions inside a finite interval. Inverse Probl Sci Eng, 2019, 27(3): 407-421
[9]	郑召文, 李昆, 支运芳. 边界条件含有谱参数的非自伴不连续 Sturm-Liouville 算子的逆谱问题. 中国科学: 数学, 2024, 54(7): 989-1008
	Zheng Z W, Li K, Zhi Y F. Inverse spectral problems of the non-selfadjoint discontinuous Sturm-Liouville operator with boundary conditions dependent on spectral parameters. Sci China Math, 2024, 54(7): 989-1008
[10]	Glazman I M. An analogue of the extension theory of Hermitian operators and a non-symmetric one-dimensional boundary problem on a half-axis. Dokl Akad Nauk SSSR, 1957, 115(2): 214-216
[11]	Galindo A. On the existence of $J$-selfadjoint extensions of $J$-symmetric operators with adjoint. Comm Pure Appl Math, 1963, 15(4): 423-425
[12]	Zhikhar N A. The theory of extensions of $J$-symmetric operators. Ukrain Mat Zh, 1959, 11: 352-364
[13]	Knowles I. On the boundary conditions characterizing $J$-selfadjoint extensions of $J$-symmetric operators. J Differ Equ, 1981, 40(2): 193-216
[14]	Race D. The theory of $J$-selfadjoint extensions of $J$-symmetric operators. J Differ Equ, 1985, 57(2): 258-274
[15]	Shang Z J. On $J$-selfadjoint extensions of $J$-symmetric ordinary differential operators. J Differ Equ, 1988, 73(1): 153-177
[16]	王爱平, 孙炯. 具有内部奇异点的 $J$-对称算子的 $J$-自共轭延拓. 南京理工大学学报 (自然科学版), 2007, 31(6): 673-678
	Wang A P, Sun J. $J$-self-adjoint extensions of $J$-symmetric operators with interior singular points. Journal of Nanjing University of Science and Technology, 2007, 31(6): 673-678
[17]	Bao Q L, Hao X L, Sun J. Characterization of self-adjoint domains for two-interval even order singular $C$-symmetric differential operators in direct sum spaces. Discrete Dyn Nat Soc, 2019, 2019(1): 1-12
[18]	Li J, Xu M Z. $J$-selfadjointness of a class of high-order differential operators with transmission conditions. Front Math, 2023, 17(6): 1025-1035
[19]	王忠, 傅守忠. 线性算子谱理论及其应用. 北京: 科学出版社, 2013 Wang Z, Fu S Z. Spectral Theory of Linear Operators and Its Applications. Beijing: Science Press, 2013
[20]	Zettl A. Sturm-Liouville Theory. New York: American Mathematical Society, 2005
[21]	刘景麟. 常微分算子谱论. 北京: 科学出版社, 2009
	Liu J L. Ordinary Differential Operator Spectrum Theory. Beijing: Science Press, 2009